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On Solvability of Higher Degree Polynomial Equations Samuel Bonaya Buya Mathematics/Physics Teacher at Ngao Girls, Secondary School, Kenya

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On Solvability of Higher Degree Polynomial Equations Samuel Bonaya Buya Mathematics/Physics Teacher at Ngao Girls, Secondary School, Kenya Research & Reviews: Journal of Applied Science and Innovations On Solvability of Higher Degree Polynomial Equations Samuel Bonaya Buya Mathematics/Physics Teacher at Ngao girls, Secondary School, Kenya Research Article Received date: 23/06/2017 ABSTRACT Accepted date: 28/07/2017 I present methods that can be used to obtain algebraic solution Published date: 07/08/2017 of polynomial equations of degree five and above. In this contribution I look into methods by which higher degree polynomial equations can *For Correspondence be factorized to obtain lower degree solvable auxiliary equations. The factorization method has been used to successfully solve quartic Samuel BB, Mathematics/Physics teacher equations. A careful selection of the appropriate factorized form can bear at Ngao girls, Secondary School, Kenya much fruit in solving higher degree polynomial. Ehrenfried Walter von Tschirnhaus (1651-1708) invented the Tschirnhaus transformation. The E-mail: [email protected] Swedish algebraist Erland Bring (1736-1798) showed by a Tschirnhaus transformation that the general quintic equation can be transformed to the trinomial form. The English mathematician George Jerrard Keywords: Algebraic solution of Quintic, Sextic and septic equations; General (1804-1863) generalized this result to higher degree polynomial. The solution of polynomial equations; Review possibility of solvability of higher degree polynomials would pave way of Galois theory; Abel-Ruffini impossibility for transformations that can reduce higher degree polynomials to their theorem; Lagrange and Galois resolvents trinomial form. The Newton Identity relates the roots of polynomials with their coefficients. It is possible to introduce an instantiation of this formula where a root of polynomial is correlated to its coefficient. This is in order to facilitate easy reduction of polynomials to lower degrees for solvability. Once a polynomial is reduced to solvable lower degree forms and there consequent roots it is possible covert it as a root of the degree of original polynomial. The paper will seek to address briefly on the things highlighted in this abstract. Solvability of higher degree polynomials will of necessity call for a re-examination of the Abel-Ruffini impossibility theorem and the Galois Theory at large. INTRODUCTION Background and Literature Review Several mathematicians have tried to obtain a radical solution of the quintic equation. Up to the past century no one succeeded to come up with its general algebraic solution. Joseph Louis Lagrange (1736-1814) wrote a book [1] in which he examined previous attempts to solve the quintic equation. Lagrange [2] introduced the concept of the Lagrange Resolvent and noted that the Resolvent worked for cubic and quartic equations but failed to achieve the desirable results for higher degree equations. In his contribution it is noted that the degree of the Resolvent of a polynomial is equal to the order of the sn. This means any attempt of solving a polynomial of degree 5 would lead to an equation of degree 120 (which is also the order of s5) In math a Resolvent is an equation upon whose solution the solution of a problem depends. For the purpose of this paper I will touch on a few highlights of the Lagrange Resolvent. For an nth degree polynomial the Lagrange Resolvent is defined as = Rx(ωω) = in i−1 Σi=1 i 1 where x_i, i=1,. , n, are the roots of the equation and ω is an nth root of unity. For a cubic equation it means by the method of Lagrange resolvents, 2 zxx=++12ωω x 3 RRJASI | Volume 1 | Issue 2 | July, 2017 7 Research & Reviews: Journal of Applied Science and Innovations Where xi are the roots of the cubic equation and ω is the cube root of unity. By permuting these roots into each other we get six different values of z. The six different permutations are: 2 z112=++ xxωω x 3 zz21= ω 2 zz31= ω 2 z4=++ xx 13ωω x 2 zz54= ω 2 zz64= ω The equation ( xz−1223456)( x −−−−−= z)( xz)( xz)( xz)( xz) 0 is then formed. We then find out that 33 ( xz−122)( x − z)( xz − 3)( xz −=− 4) x z 1 And 33 ( xz−456)( xz −)( xz −=−) x z 4 So that 3333 6 33333 ( xz−1223456)( x −−−−−=−−=−++ z)( xz)( xz)( xz)( xz) ( x z 1)( x z 4) x( z 1414 zx) zz The above equation is quadratic in x3. The importance of the Lagrange resolvent is its ability to solve the cubic and even quartic equations. However if a similar construction is made with a quartic equation it would yield a solvable equation of degree 24. With the quintic equation however the Lagrange Resolvent would yield an unsolvable degree 120 polynomial equation. The Lagrange resolvent failed to provide a way forward to the solution of higher degree polynomial equations. In 1799 Paolo Ruffini provided an incomplete proof of the impossibility of solving quintic and higher degree equations. In 1826 Niels Henrik Abel [3] provided a proof to show the impossibility of solving general degree five equations and above. Evariste Galois (1832) [4] constructed the Galois resolvent that is, if we have a polynomial with coefficients in a field F then the resolvent: in= = is a Galois resolvent if one gets n! different functions when permuting the roots with each v∑ axiiaa12, . ., an ∈ R i=1 other. Pv=−− vv vv. vv − The Galois resolvent is then used to form the polynomial ( ) ( 12)( ) ( n ) where vi are symmetric functions of the roots of the polynomial. With His group theory Galois made the observation that because it is impossible to have a chain of groups for permutation groups of sn ≥ 5 then it is impossible to solve algebraically equations of degree five and above. Galois Theory identified criteria for obtaining solvable cases. Bring [5] and Jerrard [6] how the general quintic equation can be reduced to trinomial form There is the Dummit proof theorem [7] which states that the irreducible quantic f( x) =+ x5 px 32 + qx ++ rx s is solvable in radicals if and only if the sextic equation x6++8 ax 5 40 a 24 x + 160 a 33 x + 400 a 42 x +( 512 a 5 − 3125) x + 256 a6 − 9375 ab4 = 0 has a rational root. If this is the case the sextic factors into the product of a linear polynomial (x - θ) and an irreducible quintic g(x). I will show later in this paper how this sextic can be reduced to linear and quadratic factors. Edward Thabo Motlotle [8], in his 2011 master’s thesis managed to present a formula for solving the Bring-Jerrard quintic equation using the Newton’s sum formula. In his contribution Motlotle convincingly argued that Abel’s impossibility proof has been misconstrued by many as meaning that no general algebraic solution of the quintic equation is attainable. He showed that such a formula in unattainable only within a field of Rational numbers. He then moved on to deriving a formula. Motlotle demonstrated that the Galois group associated with the general quintic equation is solvable over algebraic numbers. RRJASI | Volume 1 | Issue 2 | July, 2017 8 Research & Reviews: Journal of Applied Science and Innovations Motlotle identified a pattern that flows from the work of del Ferro (1465 – 1526) and Ferrari (1522 – 1565) in their approach to the cubic equation. The pattern he identified was of the form: rn − KL −=0 Statement of the Problem Could it be possible that the Galois Theory is incomplete in its treatment of higher degree polynomials? What about the Lagrange and Galois resolvents, are they all that the mathematics principles can provide in the solution of higher degree algebraic equations? Is it possible to come up with factorized forms that can be used to solve higher degree polynomials? Is it possible to come up with instantiations of the Newton’s sum formula? In the most general sense, the newton’s identity is a correlation between roots and coefficients of a polynomial equation. Could it be possible to come up with a correlation connecting the coefficients of a polynomial to a single root? If such a correlation can be obtained should make it possible to factorize a degree n polynomial to linear and degree n-1 factors. Is it possible that after reducing a higher degree polynomial and obtaining solutions in terms of lower degree polynomial to covert the same in forms of roots of the original polynomial? In this paper I will seek to present methods of factorizing higher degree polynomial s to solvable forms. I will seek to present an instantiation of Newton’s identity that can be used factor linear factors from out polynomial equations. I will seek to show that the sextic equation of Dummit’s proof theorem is reducible to quintic and linear factors. OBJECTIVES The main objective of this paper is to show that algebraic equations of degree five and above are solvable algebraically. Specifically I will seek to present a method by which a degree n polynomial can be factored to a linear factor and degree n-1 polynomial. I will seek to present solvable factorized forms of polynomial equations. I will seek to present a method of converting back a root to nth root form where the polynomial has been reduced to lower forms and roots have obtained in lower forms. METHODOLOGY Approach and Justification Abstract algebra asserts the impossibility of solving a general algebraic equation of degree n ≥ 5 because the Alternating group is simple that is it does not permit normal subgroups. In the paper “Solution of Solvable Irreducible Quintic Equations, without the aid of a Resolvent Sextic” [9], George Paxton Young asserts the irreducibility of the Bring-Jerrard quintic equation. G.P. Young, on the basis of irreducibility of the quintic equation in the trinomial form, argues further that it cannot be solved algebraically except in particular cases.
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